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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution:</dfn> The characteristic equation is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
r^3-3 r-2=0.
\end{equation*}
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<p class="continuation">The root to this equation is obtained as</p>
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\begin{equation*}
\begin{aligned}
0=r^3-3 r-2=r^3-r-(2 r+2)&amp;=r(r-1)(r+1)-2 (r+1)\\
&amp;=(r+1)(r^2-r-2)=(r+1)^2(r-2)\\
&amp; \to r=2 ~\textrm{or}~ -1 ~(\textrm{repeated}).
\end{aligned}
\end{equation*}
</div>
<p class="continuation">So <span class="process-math">\(e^{2 x}, e^{-x}, x e^{-x}\)</span> are a fundamental set of solutions and the general solution is</p>
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\begin{equation*}
y=C_1 e^{2 x}+C_2 e^{-x}+C_3 x e^{-x}.
\end{equation*}
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<span class="incontext"><a href="sec4_2.html#p-161" class="internal">in-context</a></span>
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